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1: 27.13 Functions
Jacobi (1829) notes that r 2 ( n ) is the coefficient of x n in the square of the theta function ϑ ( x ) : …(In §20.2(i), ϑ ( x ) is denoted by θ 3 ( 0 , x ) .) …
27.13.5 ( ϑ ( x ) ) 2 = 1 + n = 1 r 2 ( n ) x n .
27.13.6 ( ϑ ( x ) ) 2 = 1 + 4 n = 1 ( δ 1 ( n ) - δ 3 ( n ) ) x n ,
Mordell (1917) notes that r k ( n ) is the coefficient of x n in the power-series expansion of the k th power of the series for ϑ ( x ) . …
2: 4.42 Solution of Triangles
4.42.1 sin A = a c = 1 csc A ,
4.42.2 cos A = b c = 1 sec A ,
4.42.3 tan A = a b = 1 cot A .
4.42.5 c 2 = a 2 + b 2 - 2 a b cos C ,
4.42.6 a = b cos C + c cos B
3: 15 Hypergeometric Function
Chapter 15 Hypergeometric Function
4: 13 Confluent Hypergeometric Functions
Chapter 13 Confluent Hypergeometric Functions
5: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
10.46.3 E a , b ( z ) = k = 0 z k Γ ( a k + b ) , a > 0 .
This reference includes exponentially-improved asymptotic expansions for E a , b ( z ) when | z | , together with a smooth interpretation of Stokes phenomena. …
6: 8.17 Incomplete Beta Functions
where, as in §5.12, B ( a , b ) denotes the beta function: …
8.17.4 I x ( a , b ) = 1 - I 1 - x ( b , a ) .
For the hypergeometric function F ( a , b ; c ; z ) see §15.2(i). …
8.17.13 ( a + b ) I x ( a , b ) = a I x ( a + 1 , b ) + b I x ( a , b + 1 ) ,
8.17.20 I x ( a , b ) = I x ( a + 1 , b ) + x a ( x ) b a B ( a , b ) ,
7: 27.20 Methods of Computation: Other Number-Theoretic Functions
To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). … To compute a particular value p ( n ) it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
8: Peter L. Walker
Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …
9: 21.8 Abelian Functions
An Abelian function is a 2 g -fold periodic, meromorphic function of g complex variables. …For every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
10: Mourad E. H. Ismail
 Garvan), Kluwer Academic Publishers, 2001; and Theory and Applications of Special FunctionsA volume dedicated to Mizan Rahman (with E. …